algebraic manipulation exercises · 1 manipulating algebraic expressions contents general notes 4...
TRANSCRIPT
1 MANIPULATING ALGEBRAIC EXPRESSIONS
CONTENTS
GENERAL NOTES 4
SECTION 1 – EXERCISES 1 - 4 – MULTIPLYING, DIVIDING & COLLECTING
TERMS 5
SECTION 2 – EXERCISES 5 - 9 – WORKING WITH BRACKETS #1 5
SECTION 3 – EXERCISES 10 – 13 – MULTIPLYING GROUPS OF BRACKETS 5
SECTION 4 – EXERCISES 14 – 21 – FACTORISING QUADRATICS 5
SECTION 5 – EXERCISES 22 – 23 – COMPLETING THE SQUARE 6
SECTION 6 – EXERCISES 24 - 27 – ALGEBRAIC FRACTIONS 6
EXERCISE 1 – MULTIPLYING EXPRESSIONS 7
Exercise 1 – Multiplying expressions – Answers 8
EXERCISE 2 – DIVIDING EXPRESSIONS 9
Exercise 2 – Dividing expressions – Answers 10
EXERCISE 3 – TERMS 11
Exercise 3 – Terms – Answers 12
EXERCISE 4 – MIXED EXERCISE ON MULTIPLYING, DIVIDING & COLLECTING
LIKE TERMS 13
Exercise 4 – Mixed exercise on multiplying, dividing & collecting like terms – Answers 14
EXERCISE 5 – MULTIPLYING A BRACKET BY A SINGLE TERM 15
Exercise 5 – Multiplying brackets by a single term – Answers 16
EXERCISE 6 – MULTIPLYING A BRACKET BY A SINGLE TERM (SHORT) 17
EXERCISE 7 - FACTORISING SIMPLE EXPRESSIONS 17
Exercise 6 – Multiplying brackets by a single term – Answers 18
Exercise 7 – Factorising simple expressions – Answers 18
2 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 8* – FACTORISING BY GROUPING 19
EXERCISE 9 - SIMPLE FACTORISING (MIXED EXERCISE) 19
Exercise 8* – Factorising by grouping – Answers 20
Exercise 9 – Simple factorising (mixed exercise) – Answers 20
EXERCISE 10 – MULTIPLYING PAIRS OF BRACKETS (SIGNS THE SAME) 21
Exercise 10 – Multiplying pairs of brackets #1 – Answers 22
EXERCISE 11 – MULTIPLYING PAIRS OF BRACKETS (DIFFERENT SIGNS) 23
Exercise 11 – Multiplying pairs of brackets #2 – Answers 24
EXERCISE 12 – MULTIPLYING BRACKETS (MORE COMPLEX EXAMPLES) 25
EXERCISE 13 – MULTIPLYING 3 LINEAR FACTORS 25
Exercise 12 – Multiplying pairs of brackets #3 – Answers 26
Exercise 13 – Multiplying 3 Linear Factors – Answers 26
EXERCISE 14 – FACTORISING QUADRATIC EXPRESSIONS #1 (a = 1 & b, c > 0) 27
EXERCISE 15 – FACTORISING QUADRATICS #2 (a = 1, b < 0 & c > 0) 27
Exercise 14 – Factorising quadratic expressions (a = 1 & b, c > 0) – Answers 28
Exercise 15 – Factorising quadratics #2 (a = 1, b < 0, c > 0) – Answers 28
EXERCISE 16 – FACTORISING QUADRATICS #3 (a = 1, c < 0) 29
EXERCISE 17 – FACTORISING QUADRATICS #4 (MIXED EXERCISE) 29
Exercise 16 – Factorising quadratics #3 (a = 1, c > 0) – Answers 30
Exercise 17 – Factorising quadratics #4 (a = 1, c < 0) – Answers 30
EXERCISE 18 – FACTORISING QUADRATICS #5 (a > 1) 31
EXERCISE 19 – FACTORISING QUADRATICS #6 (a > 1, b < 0, c > 0) 31
Exercise 18 – Factorising quadratics #5 (a > 1, c > 0) – Answers 32
Exercise 19 – Factorising quadratics #6 (a > 1, c < 0) – Answers 32
EXERCISE 20 – FACTORISING QUADRATICS #7 (a > 1, c < 0) 33
EXERCISE 21* – FACTORISING QUADRATICS #8 (MIXED EX WHERE a > 1) 33
Exercise 20 – Factorising quadratics #6 (a > 1, signs different) – Answers 34
Exercise 21* – Factorising quadratics #7 (Mixed exercise) – Answers 34
3 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 22 – COMPLETING THE SQUARE (a = 1) 35
EXERCISE 23* – COMPLETING THE SQUARE #2 (a ≠ 1) 35
Exercise 22 – Completing the square (a = 1) – Answers 36
Exercise 23* – Completing the square (a > 1) – Answers 36
EXERCISE 24 – SIMPLIFYING ALGEBRAIC FRACTIONS 37
Exercise 24 – Simplifying algebraic fractions – Answers 38
EXERCISE 25 – ADDING & SUBTRACTING ALGEBRAIC FRACTIONS 39
Exercise 25 – Adding and subtracting algebraic fractions – Answers 40
EXERCISE 26 – MULTIPLYING & DIVIDING ALGEBRAIC FRACTIONS 41
Exercise 26 – Multiplying and dividing algebraic fractions 42
EXERCISE 27 – MIXED ALGEBRAIC FRACTIONS PROBLEMS 43
Exercise 27 – Mixed exercise on Algebraic Fractions – Answers 44
4 MANIPULATING ALGEBRAIC EXPRESSIONS
GENERAL NOTES
These materials are organised in (roughly) increasing levels of difficulty. Though the order can be
changed if your scheme of work, for example, covers multiplying binomials before simple
factorisation. However, the order in which the techniques appear should not be changed too much
as many techniques depend on prior ones.
The exercises are likewise designed to be done more or less in the order presented, as, in general
they become more difficult as they progress. If you are doing revision work you may want to
concentrate on the mixed exercises which appear at various points, though for introductory work,
the examples are often grouped so that patterns can be noticed by the students. An example of this
might be where factorising 2 6x x is followed by 2 6x x .
Throughout the booklet, x has been substituted for other letters, so that students don’t say things
like “I can do it with x but not with a.” and other similar gripes. The letters have been made the
same if a pattern point is being made, as above.
The answers are given after each exercise (or each pair in some cases) so that students can quickly
check their answers after each attempt. This is to avoid the unwelcome outcome where a student
will practise the same error over and over again, and become extremely adept at making that error.
(Unlearning is much harder than learning!)
There are as many different ways of laying out mathematical calculations as there are teachers, so
I have intentionally not suggested methods, though I may add these later in the teacher notes and
lesson plans, so that you can encourage your students to do it “the right way”.
It is important to note that these exercises are but one resource and it is probable that you will
want to use other resources in addition, or turn some of the examples into Tarsia jigsaws. The idea
behind these exercises is to give a lot of examples to use, as most text book don’t provide enough
and many online exercises are created randomly using some algorithm or other and so do not
point up the patterns. How you use these exercises is entirely up to you. In addition, the mixed
exercises make very good consolidation or assessment materials.
Most of the exercises are fairly self-explanatory, but I have added a few notes below on some of
them.
Finally, please do not give this pack in its entirety to your year 7s. They will only become terrified
unnecessarily and eventually join the ranks of adults who hate and “can’t do” mathematics.
5 MANIPULATING ALGEBRAIC EXPRESSIONS
SECTION 1 – EXERCISES 1 - 4 – MULTIPLYING , DIVIDING & COLLECTING TERMS
The first section contains exercises for practice in the four basic operations (× ÷ + –) with
algebraic expressions. It is worth noting that students should get into the habit of re-writing any
algebraic term with the letters in alphabetical order, and (of course) the numerical digit(s) before
the variable(s). Technically, it is not incorrect to write them differently (cab as opposed to abc)
but it makes later work much easier if alphabetical order is always used.
I would always encourage the students to combine terms using three distinct steps:
1. Sign,
2. Coefficient,
3. Letters.
The internal order is unimportant, but three simple steps are always better than one confused step.
Those not familiar with this idea, please refer to “How to Solve it” by Georg Polya, which will
teach you how to be a better mathematics teacher, like almost no other.
SECTION 2 – EXERCISES 5 - 9 – WORKING WITH BRACKETS #1
This section deals with single brackets for the most part, though later questions do look at how
simple factorisation can be applied to grouped brackets. These questions and/or exercises are
marked with an asterisk (*) and can be used to stretch the more able students in your care.
SECTION 3 – EXERCISES 10 – 13 – MULTIPLYING GROUPS OF BRACKETS
This section deals with groups of binomials. Again the examples have been grouped together. So
in exercise 10 Q1, a signs are +, in Q2 all signs are –. Don’t neglect Q3 which consists of squares
written as a single binomial raised to the power 2.
Again, I have not suggested a specific method, though in my experience, and at the risk of losing
my readers, FOIL is the best.
In exercise 13, we introduce multiplication of 3 linear factors.
SECTION 4 – EXERCISES 14 – 21 – FACTORISING QUADRATICS
These exercises are also in order of difficulty and grouped according to type. Many GCSE (or
equivalent) students will not need to go beyond exercise 16, after which a > 1. As with previous
exercises there are patterns to be spotted amongst adjacent questions and students should be
encouraged to look out for patterns as these work through them.
6 MANIPULATING ALGEBRAIC EXPRESSIONS
SECTION 5 – EXERCISES 22 – 23 – COMPLETING THE SQUARE
As we are now well into the year 11 (top set) territory, many students will not need to cover these
exercises, though I use them to set would be ‘A’ level students useful material to use over the
summer at the end of year 11.
SECTION 6 – EXERCISES 24 - 27 – ALGEBRAIC FRACTIONS
The final exercises involve simplification and combination of algebraic fractions. I find that ‘A’
level students who struggle with the course, often have major problems with these techniques, so
these also are good for the summer after year 11, but are also useful for students doing additional
mathematics courses.
The mantra for all of these problems is “factorise first”. If students can get this idea, they will go
on to mathematical success.
7 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 1 – MULTIPLYING EXPRESSIONS
This exercise gives practice at multiplying numbers and letters together to produce simplified
algebraic terms. Q1 – 6 only use positive numbers.
Q7 introduces directed (negative) numbers.
Q8 introduces indices.
Simplify these expressions.
1. a 3 × d a 2 × p c 5 × q
d a × 4 e w × 5 f y × 8
2. a c × d b b × a c z × x
d p × q e r × s f m × n
3. a 2a × b b 3p × q c 6s × t
d i × 4j e n × 7m f u × 4v
4. a 2p × 3 b 5q × 3 c 6c × 8
d 9 × 7t e 11 × 3g f 6 × 10j
5. a 2x × 3y b 4p × 2q c 7i × 5j
d 4f × 3g e 7a × 8b f 9d × 6e
6. a 3f × g × r b u × 4v × w c k × h × 6i
d t × 2r × 3s e a × 3c × 8b f 5s × 5r × 3p
g 6h × 2g × 4i h 3n × 7p × 2m i 2g × 5f × 6e
j 5fg × 7e × 2c k 4mn × 8jk × 5pq l 10x × 10z × 3y × 2w
7. a –2 × p b q × –5 c –2m × 3n
d –5j × –k e –3g × –5h f 9s × –8t
g x × –y × 3z h –5pq × –6ab × 4cd i –q × 8rs × –4t
j –4bc × –6ad × –2ef k 9rqs × –4b × –a l –7de × –8fg × –2h
8. a e × e b t × t c a × a
d 4q × q e 3r × r f 7c × c
g s × 3s h 5r × 6r i 4x × 3x
j 2f × 3f k –y × y × 8y l g × –5g × 2g
9. a z × –6z × 3z b –6t × –5t2 c 2i × 4i3
d 5t2 × –3t × –t e p2 × –3p × –5p f –5v2 × –7v2 × –2v
g 2x2 × –8x2 h –3q4 × –5q3 × –6q2 i –7p2q × –2pq
j –8a5b × –6ab2 k –9p3q2r × 3pqr l 6xy2z × –2x3y3z5
8 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 1 – MULTIPLYING EXPRESSIONS – ANSWERS
1. a 3d a 2p c 5q
d 4 a e 5w f 8y
2. a cd b ab c xz
d pq e rs f mn
3. a 2ab b 3pq c 6st
d 4ij e 7mn f 4uv
4. a 6p b 15q c 48c
d 63t e 33g f 60j
5. a 6xy b 8pq c 35ij
d 12fg e 56ab f 54de
6. a 3fgr b 4uvw c 6hik
d 6rst e 24abc f 75prs
g 48ghi h 42mnp i 60efg
j 70cefg k 160jkmnpq l 600wxyz
7. a –2p b –5q c –6mn
d 5jk e 15gh f –72st
g –3xyz h 120abcdpq i 32qrst
j -48abcdef k 36abqrs l –112defgh
8. a e2 b t2 c a2
d 4q2 e 3r2 f 7c2
g 3s2 h 30r2 i 12x2
j 6f 2 k –8y3 l –10g3
9. a –18z3 b 30t3 c 8i4
d 15t4 e 15p4 f –70v5
g –16x4 h –90q9 i 14p3q2
j 48a6b3 k –27p4q3r2 l –12x4y5z6
9 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 2 – DIVIDING EXPRESSIONS
This exercise gives practice at dividing numbers and letters and writing the result as a fraction.
Q4 introduces negative terms.
Q5 introduces indices as well.
Write the following expressions without the ÷ sign and simplify them where possible.
1. a 3a ÷ 3 b 4h ÷ 2 c 8p ÷ 4
d 27d ÷ 3 e 35e ÷ 7 f 48g ÷ 6
2. a a ÷ a b c ÷ c c 2x ÷ x
d 5w ÷ w e 12y ÷ 2y f 25x ÷ 5x
3. a 6s ÷ 18s b 3t ÷ 12t c 12pq ÷ 2p
d 3s ÷ 15s e 20ab ÷ 4b f 35dc ÷ 5cd
4. a 4x ÷ –2 b –5t ÷ –t c –6p ÷ 3p
d 6pq ÷ 42qp e mn ÷ 2nm f 2rt ÷ 2tr
5. a 16efg ÷ 8ef b 30hij ÷ –5hj c –6zy ÷ 5zx
d –6cd ÷ –9de e 10ab ÷ –15bc f –6xzy ÷ 14xyz
6. a g2 ÷ g2 b e2 ÷ e2 c a2 ÷ a
d 2t2 ÷ t e i2 ÷ i f y3 ÷ y2
7. a 2b3 ÷ 2b b 3c3 ÷ 2c2 c u3 ÷ 2u2
d –m5 ÷ m3 e 5j3 ÷ –j2 f –6y3 ÷ –3y
8. a –8p4 ÷ –4p3 b –10k3 ÷ 2k c pq2 ÷ –pq
d x3y2 ÷ –xy e –m3n4 ÷ mn2 f p2q2r3 ÷ pqr
9. a 4u7v5w4 ÷ 2wvu b 24i3j3k3 ÷ –6i2j2k2 c –16a3b3c3 ÷ 8a2b2c2
d 6x5y3 ÷ 3x5y2 e 20p2q5r3 ÷ –4pq2r f –15e4f 5g6 ÷ –3e2g 4f 3
10 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 2 – DIVIDING EXPRESSIONS – ANSWERS
Write the following expressions without the ÷ sign and simplify them where possible.
1. a a b 2h c 2p
d 9d e 5e f 8g
2. a 1 b 1 c 2
d 5 e 6 f 5
3. a 13
b 14
c 6q
d 15
e 5a f 7
4. a –2x b 5 c –2
d 17
e 12
f 1
5. a 2g b –6i c 6
5
y
x
d 23
ce e 2
3ac
f 37
6. a 1 b 1 c a
d 2t e i f y
7. a b2 b 32c c
2u
d –m2 e –5j f 2y2
8. a 2p b –5k2 c -q
d –x2y e –m2n2 f pqr2
9. a 2u6v4w3 b –4ijk c -2abc
d 2y e –5pq3r2 f 5e2f 2g2
11 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 3 – TERMS
Identify the terms in the following expressions by putting a box around each one. Remember that
each term includes the sign before it. It is not necessary to simplify the expressions, but try to
identify those which are like terms.
1. a 1a b b 3d e f c 7 3 2g h i
d 4 3j k e m n m f 2 3 4p p p
2. a 2a b c b q q q c 2rs rs st
d 2 5 4uv vw wx e 7mn pq f 4 8st st
g 3 24 5b b b h 23 2 5y yz y i 3 210 3 4a a a
Box the terms in these expressions and write the following expressions as simply as possible.
3. a k + k b a + a + a c y + y – y
d 2p + p e 5m – m f 4n + 2n
g 2e + 5e h 5x – 2x i 3y – 7y
j 3z – 4z k 2a + 3b – a – 3b l 5q – 4p + 2q + p
4. a 4e + 3ef + 3e – 2ef b 3i + j – i + 4j c m + n – m + n
d 4cd + 2dc e 3ab – ba f 5xyz – 2zyx
g 2ab – 4 + 3ba – 2 h 2g – 2fg + gf – 2g i 5c – 3dc + 5cd – 2c
j 4mn – 4m + 3nm – 2m k 2r + 3s + 4t – s + r – t l 2xy + ba – 2yx – ab
5. a j2 + j2 b s2 + s c 2b2 + 3b2
d a2 – a2 e t2 – 2t2 f c2 – c – c2
g e3 – e3 + e3 h g4 + 2g4 i 2j2 – j2 + j – j
j x2 + 2x + 3x2 + 3x k v2 – 2v + v2 + 3v l 4p2 + 3p – 2p2 – p
6. a h2 – 2g2 + 2h2 – g2 b 5x – 2x2 + x2 – 2x c u + 3v – 3u + v2
d (3xy)2 + 4(yx)2 e 5(ef )3 – (3fe)3 f 2a2bc + 3bca2
g 2c3d + 3c – 3c – dc3 h 5f 2 + 2g3 + 2f 2 – 2g3 i 2x2 + x – 2 + 3x2 – 4x + 5
j 7i2j + 3i2 – 4ji2 + i2 k m2n3p – n(mn)2p l (3sq)2r + (4q2r)s2
12 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 3 – TERMS – ANSWERS
1. a +a, +b, –1 b +3d, –e, –f c +7g, +3h, –2i
d +4j, –k, +3 e +m, +n, –m f +2p, +3p, –4p
2. a +a, +b, +c, –2 b +q, –q, +q c +rs, +2rs, –st
d +2uv, +5vw, –4wx e +mn, +pq, –7 f +4rs, –st, +8
g +b3, –4b2, +b, –5 h +3y2, –2yz, +5y i +10a3, –3a2, –4a
3. a 2k b 3a c y
d 3p e 4m f 6n
g 7e h 3x i –4y
j –z k a l 7q – 3p
4. a 7e + ef b 2i + 5j c 2n
d 6cd e 2ab f 3xyz
g 5ab – 6 h –fg i 3c + 2cd
j 7mn – 6m k 3r + 2s +3t l 0
5. a 2j2 b s2 + s c 5b2
d 0 e –t2 f –c
g e3 h 3g4 i j2
j 4x2 + 5x k 2v2 + v l 2p2 + 2p
6. a 3h2 – 3g2 b 3x – x2 c v2 + 3v – 2u
d 13x2y2 e –22e3f 3 f 5a2bc
g c3d h 7f 2 i 5x2 – 3x + 3
j 4i2 + 3i2j k 0 l 13q2rs2
13 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 4 – MIXED EXERCISE ON MULTIPLYING, DIVIDING & COLLECTING LIKE TERMS
This is a consolidation exercise which could be set as a homework
Simplify the following expressions.
1. a 5 × t b s ÷ 3 c uv × –4
d q + q e m + 2m f –pr ÷ r
2. a a ÷ ab b c × –c c 4g × h
d 4d – d e 2 ÷ f f 8j – 3j
3. a k + k – 2k b 2j × 3j c 6n – 7n
d 3y2 × –3y e 4x2 ÷ 8x f 6ei × –6ei
4. a 2w ÷ 3w2 b 4c2 + 5c2 c 8z2 ÷ 24z3
d (vw)2 × 3wv e –18e3 ÷ 12e2f f 3pq + 20p2 – 3qp – (4p)2
5. a yz + 2zy + z b 3 ÷ p × –q c 2 × (a × b) + 2a × b
d 8z3 ÷ (–2z)3 e (5st)2 – (2ts)2 f d – (a – 3d)
6. a 3x + x × (2x – 3) b 2g – (4h + 2g) – g + 8h c x2 × (y + 3z) ÷ x – x × y
d x – 2 × (4 – x) e c × (d – e) – ec f pq – q × q + 2q2
14 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 4 – MIXED EXERCISE ON MULTIPLYING , DIVIDING & COLLECTING LIKE TERMS
– ANSWERS
1. a 5t b 3s c -4uv d 2q e 3m f –p
2. a 1b
b –c2 c 4gh d 3d e 2f
f 5j
3. a 0 b 6j2 c –n d -9y3 e 2x f –36e2i2
4. a 23w
b 9c2 c 13z
d 3v3w3 e 32
ef
f 4p2
5. a 3yz + z b 3q
p c 4ab d –1 e 21s2t2 f 4d – a
6. a 2x2 b 4h – g c 3xz d 3x – 8 e cd – 2ce f pq + q2
15 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 5 – MULTIPLYING A BRACKET BY A SINGLE TERM
From Q4, minus signs are introduced outside the bracket.
From Q5, indices are introduced.
In Q9 you should collect like terms and simplify your answers.
Expand these expressions by removing the brackets.
1. a 2(x + 1) b 3(b + 1) c 5(q + 2)
d 4(y + 3) e 2(a + 10) f 7(b + 3)
2. a 8(d – 1) b 7(c – 5) c 6(e – 3)
d 7(g – 2) e 9(s – 6) f 12(t – 12)
3. a 2(3q + 1) b 8(3p + 7) c 6(4g + 5)
d 3(2w – 5) e 5(7y – 1) f 6(9c – 1)
4. a –2(f + 2) b –3(n + 3) c –4(4x + 1)
d –1(2p + 1) e – (q + 3) f – (4j – 3)
5. a – (–k – 1) b –3(5 – r) c –10(4m + 1)
d –5(-p + 4) e –(x – 3) f -6(2n + 3)
6. a e(e – 1) b k(k + 2) c q(p – 1)
d a(e + f ) e h(3j + 2) f m(3n – 4)
7. a c(4d – b) b 2g(3h + 4) c 5r(5s – 6)
d p(–t – u) e 2v(–h + 5v) f 6q(4 – q)
8. a 7y(2 – 3y) b 4f(2f + 5) c 2p(p + 5)
d 5y(y + 2) e 6p(7p2 – 5q) f –s(3s + 4t)
9*. a –d(–e – 9d 2) b t2(4t2 – 3t) c –x2(3x2 + 8x)
d x2y(8x + y) e –n2(2n – 3) f –p2(3p + 4p2)
10*. a 2(b + 1) + (b + 2) b 2(w + 2) + 3(w + 3) c 3(s + 5) + 2(s + 3)
d 4(2m + 3) + 5(3m + 2) e 2(2r – 3) + 4(3r – 2) f 6(3q – 5) + 2(4q – 3)
11*. a 3(3b + 4) – 2(2b + 3) b 5(4p + 7) – 3(3p + 4) c 6(2j – 5) – (4j +1)
d 5(4q + 5) – 3(5q – 1) e 5a(a – 3) – 2a(2a – 5) f 7b(2b – 3c) – 3c(2c – 4b)
16 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 5 – MULTIPLYING BRACKETS BY A SINGLE TERM – ANSWERS
1. a 2x + 2 b 3b + 3 c 5q + 10
d 4y + 12 e 2a + 20 f 7b + 21
2. a 8d – 8 b 7c – 35 c 6e – 18
d 7g – 14 e 9s – 54 f 12t – 144
3. a 6q + 2 b 24p + 56 c 24g + 30
d 6w – 15 e 35y – 5 f 54c – 6
4. a –2f – 4 b –3n – 9 c –16x – 4
d –2p – 1 e –q – 3 f –4j + 3
5. a k + 1 b –15 + 3r c –40m – 10
d 5p – 20 e 3 – x f –12n – 18
6. a e2 – e b k2 + 2k c pq – q
d ae + af e 3hj + 2h f 3mn – 4m
7. a 4cd – bc b 6gh + 8g c 25rs – 30r
d –pt – pu e –2hv + 10v2 f 24q – 6q2
8. a 14y – 21y2 b 8f 2 + 20f c 2p2 + 10p
d 5y2 + 10y e 42p3 – 30pq f –3s2 – 4st
9*. a de + 9d 3 b 4t4 – 3t3 c –3x4 – 8x3
d 8x3y + x2y2 e –2n3 + 3n2 f –3p3 – 4p4
10*. a 3b + 4 b 5w + 13 c 5s + 21
d 23m + 32 e 16r – 14 f 26q – 36
11*. a 5b + 6 b 11p + 23 c 8j – 31
d 5q + 28 e a2 – 5a f 4b2 – 9bc – 6c2
17 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 6 – MULTIPLYING A BRACKET BY A SINGLE TERM (SHORT)
This is a consolidation exercise which could be set as a homework
Expand these expressions by removing the brackets.
1. a 5(x + 3) b 3(b – 1) c 7(q – 3)
d 4(y – 3) e 5(a + b) f 6(b – 4)
2. a d(d – 1) b c(2 + c) c e(e – 3)
d 2p(p + 5) e 5y(y + 2) f 4f(2f + 5)
g 5w(w2 + 5) h 11t(3 – 5t) i 6q(4 – q)
3. a n2(2n + 3) b p2(3p + 4) c t2(4t2 – 3t)
d x2y(8x + y) e 2abc(3a + 2b + c) f 3de(3d + e + 2)
g –(3x + 4) h –(5r – 6) i –(10 – t)
4. a 7(a + 3) + 4(a – 3) b 8(6p – 5) – 3(2p – 5) c –2r(r + 3) – (2 – r)
EXERCISE 7 - FACTORISING SIMPLE EXPRESSIONS
Q5 introduces indices for the first time.
Factorise these expressions fully.
1. a 4e + 4 b 2y + 6 c 9q + 3
d 3b + 3 e 5d + 10 f 4s + 6
2. a 3b – 3 b 2c – 10 c 6p – 6
d 2m – 4 e 6c – 3 f 10p – 8
3. a pq + 2p b 2s + st c ab – bc
d de – 2df e 3np – 9p f 3gh + 6g
4. a 4pq – 8p b 8tu + 4t c 4xy – 2x + 6xz
d 14xyz – 7xy e 20uvw – 2uv + 4vw f 15gte + 5gt – 10g
5. a r2 + 4r b a2 + 3a c y2 + 2y
d 4t2 + 6 e 2s – 4s2 f 6q2 – 4q
g 2b2 + 6b h 14e3 – 21e i 15g2 + 9g
6*. a 3m2 + 9m b 4p3 + 9p2 c –2x2 + 4xy
d –6tu2 – 8t e –2h – h2 f 6rs2t – 9rs
g 6t2u – 8tu2 h 18s3t2 – 12st i 14a2b + 21ab
j –15pq2 + 10qp k 5s – 10s2 – 15s4 l 7uv – 7u2v
m ( p2q)2 – p2q n (3a2b)2 + 18a3b2 o 6xy2z – 12x2yz + 4xyz2
18 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 6 – MULTIPLYING BRACKETS BY A SINGLE TERM – ANSWERS
1. a 5x + 15 b 3b –3 c 7q – 21
d 4y – 12 e 5a + 5b f 6b – 24
2. a d2 – d b 2c + c2 c e2 – 3e
d 2p2 + 10p e 5y2 + 10y f 8f 2 + 20f
g 5w3 + 25w h 33t – 55t2 i 24q – 6q2
3 a 2n3 + 3n2 b 3p3 + 4p2 c 4t4 – 3t3
d 8x3y + x2y2 e 6a2bc + 4ab2c + 2abc2 f 9d 2e + 3de2 + 6de
g –3x – 4 h –5r + 6 i –10 + t
4. a 11a + 9 b 42p – 25 c –r – 8
EXERCISE 7 – FACTORISING SIMPLE EXPRESSIONS – ANSWERS
1. a 4(e + 1) b 2(y + 3) c 3(3q + 1)
d 3(b + 1) e 5(d + 2) f 2(2s + 3)
2. a 3(b – 1) b 2(c – 5) c 6(p – 1)
d 2(m – 2) e 3(2c – 1) f 2(5p – 4)
3. a p(q + 2) b s(2 + t) c b(a – c)
d d(e – 2f ) e 3p(n – 3) f 3g(h + 2)
4. a 4p(q – 2) b 4t(2u + 1) c 2x(2y – 1 + 3z)
d 7xy(2z – 1) e 2v(10uw – u + 2w) f 5g(3et + t – 2)
5. a r(r + 4) b a(a + 3) c y(y + 2)
d 2(2t2 + 3) e 2s(1 – 2s) f 2q(3q – 2)
g 2b(b + 3) h 7e(2e2 – 3) i 3g(5g + 3)
6*. a 3m(m + 3) b p2(4p + 9) c 2x(2y – x)
d –2t(3u2 + 4) e –h(2 + h) f 3rs(2st – 3)
g 2tu(3t – 4u) h 6st(3s2t – 2) i 7ab(2a + 3)
j 5pq(2 – 3q) k 5s(1 – 2s – 3s3) l 7uv(1 – u)
m p2q(p2q – 1) n 9a3b2(a + 2) o 2xyz(3y – 6x + 2z)
19 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 8* – FACTORISING BY GROUPING
Factorise these expressions fully.
1. a 2(a + b) + (a + b) b 3(c – 2d) + e(c – 2d) c g(3m – n) – (3m – n)
d 6( p + 3) – 2( p + 3) e 3(2t + 3) + t2(2t + 3) f (3st + t2) + 4(3s + t)
g 4(5x + 1)2 – 2(5x + 1) h (m + p)2 + (m + p) i (x + y)2 – 3(x + y)
2. a (q + 2r) + 3(q + 2r)2 b (x2 + y)3 – 4(x2 + y)2 c (3m – 2n)2 – (3m – 2n)3
d (a – b) – c(a – b) e 2(m2 – n) – 3n(n – m2) f (x – 2y) – x(2y – x)
g a(2b – c) + 3(c – 2b) h (i – k)2 – (k – i) i (2x – y)2 – (2y – 4x)
3. a xy + x + y + 1 b de + 3e + df + 3f c abc – 4ab + 3c – 12
d p2q + pq +2p + 2 e 3m2 + mp + 6mn + 2np f i2jk – ij2 + ik2 – jk
EXERCISE 9 - SIMPLE FACTORISING (MIXED EXERCISE)
Factorise these expressions fully.
1. a 12t + 4 b 6x – 3 c 5s – 5
d ab – b e dc2 – c f 2d + 5de
g 3 – 6g2 h 10rs + 2s i 15pq + 10q2
2. a 8x2 – 4xy b 15p3 – 10p c 7rst + 7r
d (2b2)3 + b5 e 6abc + 9ab – 15bc f 3ab2 – 6ab + 9b
3. a 2πrh + πr2 b 2 32 1
3 3r r c –st – 3t
d –4d 2 – 6de e a(b + c) + d(b + c) f 2x(x2 – 3) – 3(x2 – 3)
g t(1 – t) – (t – 1) h (g + h)3 – (g + h)2 i 6(p – 3q) – 4(6q – 2p)
20 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 8* – FACTORISING BY GROUPING – ANSWERS
1. a 3(a + b) b (3 + e)(c – 2d) c (g – 1)(3m – n)
d 4(p + 3) e (3 + t2)(2t + 3) f (t + 4)(3s + t)
g 2(5x + 1)(10x + 1) h (m + p)(m + p + 1) i (x + y)(x + y – 3)
2. a (q + 2r)(1 + 3q + 6r) b (x2 + y)2(x2 + y – 4) c (3m – 2n)2(1 – 3m + 2n)
d (a – b)(1 – c) e (2 + 3n)(m2 – n) f (1 + x)(x – 2y)
g (a – 3)(2b – c) h (i – k)(i – k + 1) i (2x – y)(2x – y + 2)
3. a (x + 1)(y + 1) b (e + f )(d + 3) c (c – 4)(ab + 3)
d (p + 1)(pq + 2) e (m + 2n)(3m + p) f (ij + k)(ik – j)
EXERCISE 9 – SIMPLE FACTORISING (MIXED EXERCISE) – ANSWERS
1. a 4(3t + 1) b 3(2x – 1) c 5(s – 1)
d b(a – 1) e c(cd – 1) f d(2 + 5e)
g 3(1 – 2g2) h 2s(5r + 1) i 5q(3p + 2q)
2. a 4x(2x – y) b 5p(3p2 – 2) c 7r(st + 1)
d b5(8b + 1) e 3b(2ac + 3a – 5c) f 3b(ab – 2a + 3)
3. a πr(2h + r) b 2
23
rr
c –t(s + 3)
d –2d(2d + 3e) e (a + d)(b + c) f (2x – 3)(x2 – 3)
g (1 – t)(1 + t) h (g + h)2(g + h – 1) i 14(p – 3q)
21 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 10 – MULTIPLYING PAIRS OF BRACKETS (SIGNS THE SAME)
Hint: The parts of questions 1 and 2 are very similar. See if you can spot the pattern.
Q4 is a mixed of the types we have met in Q1–3
Multiply out the following pairs of brackets and then simplify your answer if possible.
1. a 1 2f f b 1 3a a c 2 3m m
d 2 4r r e 2 5g g f 3 4b b
g 3 5h h h 4 5n n i 6 2v v
j 7 3c c k 6 4s s l 8 3p p
m 5 8d d n 8 9e e o 7 9i i
2. a 1 2x x b 1 3i i c 2 3z z
d 2 4j j e 2 5d d f 3 4t t
g 3 5q q h 4 5w w i 6 2y y
j 7 3u u k 6 4k k l 8 3e e
m 5 8a a n 8 9b b o 7 9g g
3. a 2
3f b 2
3k c 2
2a
d 2
2q e 2
5v f 2
5g
g 2
7b h 2
7m i 2
6r
j 2
6h k 2
9z l 2
9w
4*. a 3 7a a b 2
3x c 3 9y y
d 6 5b b e 10 9w w f 11 9s s
g 2
m n h 2
a b i 3 3t t
j 2 – – 3r r k 3 3q q l p q q p
22 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 10 – MULTIPLYING PAIRS OF BRACKETS #1 – ANSWERS
1. a f 2 + 3f + 2 b a2 + 4a + 3 c m2 + 5m + 6
d r2 + 6r + 8 e g2 + 7g + 10 f b2 + 7b + 12
g h2 + 8h + 15 h n 2 + 9n + 20 i v2 + 8v + 12
j c2 +10c + 21 k s2 + 10s + 24 l p2 + 11p + 24
m d2 + 13d + 40 n e2 + 17e + 72 o i2 + 16i + 63
2. a x2 – 3x + 2 b i2 – 4i + 3 c z2 – 5z + 6
d j2 – 6j + 8 e d 2 – 7d + 10 f t2 – 7t + 12
g q2 – 8q + 15 h w2 – 9w + 20 i y2 – 8y + 12
j u2 – 10u + 21 k k2 – 10k + 24 l e2 – 11e + 24
m a2 – 13a + 40 n b2 – 17b + 72 o g2 – 16g + 63
3. a f 2 + 6f + 9 b k2 – 6k + 9 c a2 + 4a + 4
d q2 – 4q + 4 e v2 + 10v + 25 f g2 – 10g + 25
g b2 + 14b + 49 h m2 – 14m + 49 i r2 + 12r + 36
j h2 – 12h + 36 k z2 + 18z + 81 l w2 – 18w + 81
4*. a a2 + 10a + 21 b x2 – 6x + 9 c y2 – 12y + 27
d b2 + 11b + 30 e w2 – 19w + 90 f s2 + 20s + 99
g m2 + 2mn + n2 h a2 – 2ab + b2 i t2 + 6t + 9
j –r2 + 5r – 6 k –q2 + 6q – 9 l –p2 + 2pq – q2
23 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 11 – MULTIPLYING PAIRS OF BRACKETS (DIFFERENT SIGNS)
Q1 has been organised in pairs. Can you find a pattern in the questions and answers.
Q4 introduces difference of two squares. Try to spot the pattern in the questions and answers.
Q5 has slightly more complex examples.
Multiply out the following pairs of brackets and then simplify your answer if possible.
1. a 1 2e e b 1 2e e c 1 3q q
d 1 3q q e 2 4u u f 2 4u u
g 5 3r r h 5 3r r i 6 2k k
j 6 2k k k 7 5b b l 7 5b b
2. a 6 7m m b 2 8s s c 5 2g g
d 9 8a a e 10 5p p f 12 11r r
g 8 11b b h 16 5x x i 6 8z z
3. a 2 9z z b 4 7w w c 5 8n n
d 8 7t t e 9 3h h f 4 10c c
g 7 9p p h 3 7d d i 8 9i i
4. a 1 1s s b 3 3c c c 7 7n n
d 6 6x x e 11 11i i f 5 5t t
g 8 8p p h 10 10u u i 2 2d d
j 9 9y y k 4 4e e l 12 12j j
5*. a 3 3u u b b z b z c 5 5hd hd
d 4 4g g e 5 4x x f 3 5u u
g 8 6v v h 7 8r r i 5 3y y
j 10 11v v k 9 8i z l 3 4a b
24 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 11 – MULTIPLYING PAIRS OF BRACKETS #2 – ANSWERS
1 a e2 + e – 2 b j2 – j – 2 c q2 + 2q – 3
d a2 – 2a – 3 e u2 + 2u – 8 f x2 – 2x – 8
g r2 – 2r – 15 h f 2 + 2f – 15 i k2 – 4k – 12
j v2 + 4v – 12 k b2 – 2b – 35 l y2 + 2y – 35
2. a m2 – m – 42 b s2 + 6s – 16 c g2 – 3g – 10
d a2 + a – 72 e p2 + 5p - 50 f r2 – r – 132
g b2 + 3b – 88 h x2 +11x – 80 i z2 + 2z – 48
3 a z2 + 7z – 18 b w2 +3w – 28 c n2 + 3n – 40
d t2 – t – 56 e h2 – 6h – 27 f c2 + 6c – 40
g p2 + 2p – 63 h d 2 + 4d – 21 i i2 + i – 72
4. a s2 – 1 b c2 – 9 c n2 – 49
d x2 – 36 e i2 – 121 f t2 – 25
g p2 – 64 h u2 – 100 i d2 – 4
j y2 – 81 k e2 – 16 l j2 – 144
5*. a u2 – 9 b b2 – z2 c h2d 2 – 25
d g2 – 16 e –x2 – x + 20 f –u2 – 2u + 15
g –v2 + 2v + 48 h r2 + r – 56 i –y2 +2y + 15
j v2 – v – 110 k 72 +9z – 8i – iz l ab – 4a + 3b – 12
25 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 12 – MULTIPLYING BRACKETS (MORE COMPLEX EXAMPLES)
Hint: Remember to multiply out in 3 stages, signs, numbers & letters.
Multiply the following pairs of brackets and then simplify your answer if possible.
1. a 2 1 1f f b 4 3 1m m c 2 5 4w w
d 2 3 2 3n n e 3 2 2 3z z f 2
3 4k
2 a 2 7 4 5a a b 4 3 4 3d d c 3 4 1r r
d 2 3 2 4s s e 5 2 2 5t t f 5 2 1i i
3. a 2 5 3 2x x b 2 3 2 3h h c 5 3 5 3j j
d 2 3 1 2c c e 3 2 4 1y y f 3 1 2a a
4. a 2 3 2p p b 3 2 3 2q v q v c ax b ax b
d 4 4u u e 2 2 3p p f 2
ax b
EXERCISE 13 – MULTIPLYING 3 LINEAR FACTORS
1. a 1 2x x x b 1 3p p p c 3 4a a a
d 2 1b b b e 4 5d d d f 4 6q q q
2 a 1 1v v v b 1 2t t t c 5 3q q q
d 2 5m m m e 5 3z z z f 7 3r r r
3. a 2 3 4d d d b 4 2 4f f f c 5 2 1g g g
d 6 1 3k k k e 2 2 6b b b f 3 3 3j j j
4. a 1 1 1a a a b 2 2 2s s s c 1 1 2r r r
d 2 3 4n n n e 3 5 4g g g f 3 5 6h h h
5. a 3 2 3 3 2a a a b 3 2 5 3 2t t t c 6 4 1 3 2y y y
d 2 3 2 1 3 4x x x e 2 5 3 2 4 3u u u f 3 7 2 5 3c c c
*6. a 23 4 3 2 3 1p p p b 2 4 2 3 2a a a c 2 2 23 4 4 3 3 1b b b
d 2 2 21 2 3c c c e 2 2 22 3 3 2g g g
f 2 2 23 2 3 2 5m m m
26 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 12 – MULTIPLYING PAIRS OF BRACKETS #3 – ANSWERS
1. a 2f 2 + 3f + 1 b 4m2 + 7m + 3 c 2w2 + 13w + 20
d 4n2 + 12n + 9 e 6z2 + 13z + 6 f 9k2 + 24k + 16
2. a 8a2 – 38a + 35 b 16d2 – 24d + 9 c 3r2 – 7r + 4
d 4s2 – 14s + 12 e 10t2 – 29t + 10 f 5 – 7i + 2i2
3. a 6x2 – 11x – 10 b 4h2 – 9 c 25 – 9j2
d 2 – c – 6c2 e 8y2 + 10y – 3 f 3a2 + 5a – 2
4. a 25 – 9j2 b 9q2 – 4v2 c a2x2 – b2
d 16 – u2 e 4 – 4p – 3p2 f a2x2 + 2abx + b2
EXERCISE 13 – MULTIPLYING 3 LINEAR FACTORS – ANSWERS
1. a 3 23 2x x x b 3 24 3p p p c 3 27 12a a a
d 3 23 2b b b e 3 29 20d d d f 3 210 24q q q
2 a 3v v b 3 2 2t t t c 3 22 15q q q
d 3 22 10m m m e 3 22 15z z z f 3 24 21r r r
3. a 3 22 14 24d d d b 3 24 24 32f f f c
3 25 5 10g g g
d 2 318 12 6k k k e 2 324 8 2b b b f 327 3j j
4. a 3 23 3 1a a a b 3 26 12 8s s s c 3 24 5 2r r r
d 3 29 26 24n n n e 3 212 47 60g g g f 3 214 63 90h h h
5. a 3 218 39 18a a a b 3 218 33 30t t t c 3 272 30 12y y y
d 3 212 4 25 12x x x e 3 224 62 7 30u u u f 3 215 4 73 42c c c
6. a 4 3 227 54 15 18 8p p p p b 4 3 22 2 4 24a a a a c 6 4 236 11 29 12b b b
d 2 4 66 11 6c c c e 2 4 618 3 7 2g g g f 2 4 630 77 56 12m m m
27 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 14 – FACTORISING QUADRATIC EXPRESSIONS #1 (a = 1 & b, c > 0)
Hint: All of these quadratic expressions can be factorised. Try to spot patterns between examples.
Factorise the following expressions.
1. a 2 3 2x x b 2 2 1n n c 2 4 3a a
d 2 6 5p p e 2 5 4m m f
2 8 7q q
g 2 10 9s s h 2 7 6g g i 2 5 6h h
2. a 2 13 12u u b 2 8 12k k c 2 7 12r r
d 2 6 8f f e 2 8 15w w f 2 16 15v v
g 2 9 18p p h 2 19 18m m i 2 11 18a a
3. a 2 17 16k k b 2 21 20c c c 2 12 20q q
d 2 9 20b b e 2 14 24x x f 2 11 24s s
g 2 10 24t t h 2 8 16t t i 2 9 14m m
EXERCISE 15 – FACTORISING QUADRATICS #2 (a = 1, b < 0 & c > 0)
Hint: All of these quadratic expressions can be factorised. Again, take your time and try to spot
the patterns.
Factorise these expressions.
1. a 2 4 4b b b 2 2 1x x c 2 3 2s s
d 2 8 16q q e 2 5 6c c f
2 6 5p p
g 2 6 8k k h 2 5 4m m i 2 4 3a a
2. a 2 7 12w w b 2 6 9v v c 2 8 12u u
d 2 10 21r r e 2 9 18f f f 2 14 24k k
g 2 17 16t t h 2 9 20h h i 2 11 18s s
3. a 2 12 20n n b 2 11 24p p c 2 8 15m m
d 2 19 18x x e 2 16 15a a f 2 13 12q q
g 2 10 24d d h 2 25 24g g i 2 13 36e e
28 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 14 – FACTORISING QUADRATIC EXPRESSIONS (a = 1 & b, c > 0) – ANSWERS
1. a (x + 1)(x + 2) b (n + 1)(n + 1) c (a + 1)(a + 3)
d (p + 1)(p + 5) e (m + 1)(m + 4) f (q + 7)(q + 1)
g (s + 9)(s + 1) h (g + 6)(g + 1) i (h + 2)(h + 3)
2. a (u + 1)(u + 12) b (k + 2)(k + 6) c (r + 3)(r + 4)
d (f + 2)(f + 4) e (w + 3)(w + 5) f (v + 1)(v + 15)
g (p + 3)(p + 6) h (m + 1)(m + 18) i (a + 2)(a + 9)
3. a (k + 1)(k + 16) b (c + 1)(c + 20) c (q + 2)(q + 10)
d (b + 4)(b + 5) e (x + 2)(x + 12) f (s + 3)(s + 8)
g (t + 4)(t + 6) h (t + 4)(t + 4) i (m + 7)(m + 2)
EXERCISE 15 – FACTORISING QUADRATICS #2 (a = 1, b < 0, c > 0) – ANSWERS
1. a (b – 2)(b – 2) b (x – 1)(x – 1) c (s – 1)(s – 2)
d (q – 4)(q – 4) e (c – 2)(c – 3) f (p – 1)(p – 5)
g (k – 2)(k – 4) h (m – 1)(m – 4) i (a – 1)(a – 3)
2. a (w – 3)(w – 4) b (v – 3)(v – 3) c (u – 2)(u – 6)
d (r – 3)(r – 7) e (f – 3)(f – 6) f (k – 2)(k – 12)
g (t – 1)(t – 16) h (h – 4)(h – 5) i (s – 2)(s – 9)
3. a (n – 2)(n – 10) b (p – 3)(p – 8) c (m – 3)(m – 5)
d (x – 1)(x – 18) e (a – 1)(a – 15) f (q – 1)(q – 12)
g (d – 6)(d – 4) h (g – 24)(g – 1) i (e – 9)(e – 4)
29 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 16 – FACTORISING QUADRATICS #3 (a = 1, c < 0)
Hint: All of these quadratic expressions can be factorised. Once again try to spot patterns between
the examples.
Factorise the following expressions.
1. a 2 2d d b 2 2k k c 2 6r r
d 2 6x x e 2 5 6e e f 2 5 6m m
g 2 4 5w w h 2 2 8z z i 2 2 8f f
2. a 2 49u b 2 4s c 2 25t
d 2 2 15g g e 2 2 15a a f
2 15 16y y
g 2 6 16h h h 2 6 16p p i 2 3 10n n
3. a 2 19 20i i b 2 19 20v v c 2 8 20b b
d 2 15 16q q e 2 20c c f
2 20j j
g 2 5 24r r h 2 2 24k k i 2 23 24z z
EXERCISE 17 – FACTORISING QUADRATICS #4 (MIXED EXERCISE)
Factorise the following expressions.
1. a 2 2 24d d b 2 11 30j j c
2 7 18q q
d 2 5 14u u e 2 5 24w w f 2 5 24e e
g 2 15 14r r h 2 20 36z z i 2 6 27k k
2. a 2 7 10x x b 2 23 24f f c 2 9 18s s
d 2 6 7m m e 2 3 28v v f 2 2 35y y
g 2 13 30a a h 2 36n i 2 30g g
3. a 2 4 32t t b 2 8 20h h c 2 3 10b b
d 2 16p e 2 4 5c c f 2 14 32i i
g 2 42d d h 2 2 35e e i 2 56i i
30 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 16 – FACTORISING QUADRATICS #3 (a = 1, c > 0) – ANSWERS
1. a (d – 1)(d + 2) b (k – 2)(k + 1) c (r – 3)(r + 2)
d (x – 2)(x + 3) e (e – 1)(e + 6) f (m – 6)(m + 1)
g (w – 5)(w + 1) h (z – 4)(z + 2) i (f – 2)(f + 4)
2. a (u – 7)(u + 7) b (s – 2)(s + 2) c (t – 5)(t + 5)
d (g – 3)(g + 5) e (a – 5)(a + 3) f (y – 1)(y + 16)
g (h – 2)(h + 8) h (p – 8)(p + 2) i (n – 2)(n + 5)
3. a (i – 1)(i + 20) b (v – 20)(v + 1) c (b – 2)(b + 10)
d (q – 16)(q + 1) e (c – 4)(c + 5) f (j – 5)(j + 4)
g (r – 3)(r + 8) h (k – 6)(k + 4) i (z – 24)(z + 1)
EXERCISE 17 – FACTORISING QUADRATICS #4 (a = 1, c < 0) – ANSWERS
1. a (d – 6)(d + 4) b (j + 5)(j + 6) c (q – 9)(q + 2)
d (u – 2)(u+ 7) e (w – 3)(w + 8) f (e – 8)(e + 3)
g (r – 1)(r – 14) h (z + 2)(z + 18) i (k – 9)(k + 3)
2. a (x + 2)(x + 5) b (f – 1)(f + 24) c (s + 3)(s + 6)
d (m – 1)(m + 7) e (v – 4)(v + 7) f (y – 7)(y + 5)
g (a – 2)(a + 15) h (n – 6)(n + 6) i (g – 6)(g + 5)
3. a (t – 4)(t + 8) b (h – 10)(h + 2) c (b – 5)(b + 2)
d (p – 4)(p + 4) e (c – 1)(c + 5) f (i – 16)(i + 2)
g (d – 7)(d + 6) h (e + 7)(e – 5) i (i + 8)(i – 7)
31 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 18 – FACTORISING QUADRATICS #5 (a > 1)
Factorise the following quadratic expressions.
1. a 2f 2 + 3f + 1 b 3a2 + 4a + 1 c 5m2 + 6m + 1
d 6s2 + 5s + 1 e 10g2 + 7g + 1 f 8w2 + 6w + 1
g 3z2 + 10z + 3 h 6n2 + 11n + 3 i 6b2 + 13b + 6
2. a 4t2 + 7t + 3 b 3h2 + 7h + 2 c 2p2 + 5p + 2
d 6x2 + 19x + 3 e 8c2 + 14c + 3 f 6i2 + 23i + 20
g 9y2 + 12y + 4 h 4q2 + 20q + 25 i 9u2 + 24u + 16
3. a 12d 2 + 17d + 6 b 12j2 + 28j + 15 c 6r2 + 19r + 15
d 21k2 + 37k + 12 e 10e2 + 23e + 12 f 18v2 + 27v + 10
g 12i2 +47i + 40 h 35h2 + 57h + 18 i 9i2 + 27i + 20
EXERCISE 19 – FACTORISING QUADRATICS #6 (a > 1, b < 0, c > 0)
Hint: Some of the questions have a numerical factor as well as the algebraic ones.
Factorise the following quadratics fully.
1. a 2e2 – 3e + 1 b 3k2 – 4k + 1 c 5q2 – 6q + 1
d 3u2 – 5u + 2 e 6f 2 – 11f + 3 f 4v2 – 13v + 3
g 4m2 – 8m + 3 h 3r2 – 10r + 8 i 5g2 – 16g + 3
2. a 6x2 – 13x + 6 b 8a2 – 22a + 15 c 12z2 – 29z + 15
d 6s2 – 23s + 20 e 12h2 – 17h + 6 f 20w2 – 23w + 6
g 12b2 – 23b + 10 h 6n2 – 16n + 8 i 20y2 – 22y + 6
3. a 12i2 – 48i + 21 b 15p2 – 22p + 8 c 18c2 – 30c + 8
d 10p2 – 27p + 18 e 21d 2 – 34d + 8 f 20j2 – 36j + 9
g 30k2 – 47k + 7 h 24x2 – 62x + 40 i 84x2 – 188x + 80
32 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 18 – FACTORISING QUADRATICS #5 (a > 1, c > 0) – ANSWERS
1. a (2f + 1)(f + 1) b (3a + 1)(a + 1) c (5m +1 )(m + 1)
d (2s + 1)(3s+1) e (2g + 1)(5g + 1) f (4w + 1)(2w + 1)
g (3z + 1)(z + 3) h (3n + 1)(2n + 3) i (2b + 3)(3b + 2)
2. a (t + 1)(4t + 3) b (3h + 1)(h + 2) c (2p + 1)(p + 2)
d (6x + 1)(x + 3) e (2c + 3)(4c + 1) f (2i + 5)(3i + 4)
g (3y + 2)(3y + 2) h (2q + 5)(2q + 5) i (3u + 4)(3u + 4)
3. a (3d + 2)(4d + 3) b (6j + 5)(2j + 3) c (2r + 3)(3r + 5)
d (7k + 3)(3k + 4) e (5e + 4)(2e + 3) f (6v + 5)(3v + 2)
g (3i + 8)(4i + 5) h (5h + 6)(7h + 3) i (3i + 4)(3i + 5)
EXERCISE 19 – FACTORISING QUADRATICS #6 (a > 1, c < 0) – ANSWERS
1. a (2e – 1)(e – 1) b (3k – 1)(k – 1) c (5q – 1)(q – 1)
d (3u – 2)(u – 1) e (3f – 1)(2f – 3) f (4v – 1)(v – 3)
g (2m – 3)(2m – 1) h (r – 2)(3r – 4) i (5g – 1)(g – 3)
2. a (3x – 2)(2x – 3) b (4a – 5)(2a – 3) c (4z – 3)(3z – 5)
d (3s – 4)(2s – 5) e (3h – 2)(4h – 3) f (5w – 2)(4w – 3)
g (3b – 2)(4b – 5) h 2(3n – 2)(n – 2) i 2(5y – 3)(2y – 1)
3. a 3(2i – 7)(2i – 1) b (3p – 2)(5p – 4) c 2(3c – 1)(3c – 4)
d (2p – 3)(5p – 6) e (7d – 2)(3d – 4) f (10j – 3)(2j – 3)
g (5x – 7)(6x – 1) h 2(4x – 5)(3x – 4) i 4(3x – 5)(7x – 4)
33 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 20 – FACTORISING QUADRATICS #7 (a > 1, c < 0)
Hint: some of the questions have a numerical factor as well as the algebraic ones. Take the
number out first.
Factorise the following quadratics fully.
1. a 22 1a a b 22 1g g c 24 3r r
d 23 4 4v v e 22 6x x f 23 5 12h h
g 23 5 12k k h 23 10b b i 23 10x x
2. a 28 6 5b b b 24 4 15s s c 26 12m m
d 212 1 80w w e 2 910 9n n f 2 910 9c c
g 210 1 159d d h 29 4p i 218 2 41t t
3. a 232 50z b 221 2 203e e c 208 41 1 7y
d 224 1 158q q e 214 3 101u u f
224 1 244f f
g 272 98x h 212 8 15n n i 4 81t
EXERCISE 21* – FACTORISING QUADRATICS #8 (MIXED EXERCISE WHERE a > 1)
Hint: some of the questions have a numerical factor as well as the algebraic ones. Take out the
number first.
Factorise the following quadratics fully.
1. a 26 19 10g g b
24 9p c 2 112 8a a
d 221 2 203u u e 218 1 25h h f 236 6 250q q
g 29 15 36i i h 212 2 158b b i 210 2 109v v
2. a 212 1 204r r b 25 1 88j j c 220 2 63e e
d 214 9 1w w e 23 2 1y y f 212 6k k
g 24 9 9d d h 245 20s i 24 5 1x x
3. a 24 16 15z z b 28 6 1m m c 236 64e
d 216 1 102n n e 29 1 42f f f 220 2 94t t
g 212 8 15x x h 2 2 22a x abx b i 2 2 2a x b
34 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 20 – FACTORISING QUADRATICS #6 (a > 1, SIGNS DIFFERENT) – ANSWERS
1. a (2a + 1)(a – 1) b (g + 1)(2g – 1) c (r + 1)(4r – 3)
d (3v + 2)(v – 2) e (2x + 3)(x – 2) f (h + 3)(3h – 4)
g (3k + 4)(k – 3) h (b + 2)(3b – 5) i (3x + 5)(x – 2)
2. a (2b + 1)(4b – 5) b (2s + 5)(2s – 3) c (2m + 3)(3m – 4)
d 2(3w + 4)(2w – 1) e (5n + 3)(2n – 3) f (2c + 3)(5c – 3)
g (2d + 5)(5d – 3) h (3p + 2)(3p – 2) i (6t + 1)(3t – 4)
3. a 2(4z + 5)(4z – 5) b (3e + 5)(7e – 4) c 3(6y + 7)(6y + 7)
d 3(4q + 5)(2q – 1) e (2u + 5)(7u – 2) f 2(3f + 4)(4f – 3)
g 2(6x – 7)(6x + 7) h (6n – 5)(2n + 3) i (t – 3)(t + 3)(t2 + 9)
EXERCISE 21* – FACTORISING QUADRATICS #7 (MIXED EXERCISE) – ANSWERS
1. a (2g – 5)(3g – 2) b (2p + 3)(2p – 3) c (6a + 1)(2a + 1)
d (3u – 5)(7u + 4) e (6h + 1)(3h + 2) f (6q – 5)2
g 3(3i + 4)(i – 3) h (6b – 5)(2b – 3) i (2v – 5)(5v – 2)
2. a 2(6r + 5)(r – 2) b (5j + 2)(j – 4) c (5e + 2)(4e + 3)
d (7w + 1)(2w + 1) e (y + 1)(3y – 1) f (3k + 2)(4k – 3)
g (d + 3)(4d – 3) h 5(3s + 2)(3s – 2) i (4x + 1)(x + 1)
3. a (2z + 3)(2z + 5) b (2m – 1)(4m – 1) c 4(3z + 4)(3z – 4)
d 2(4n + 5)(2n – 1) e (3f – 2)2 f (2t + 3)(10t – 3)
g (6x – 5)(2x + 3) h (ax + b)2 i (ax – b)(ax + b)
35 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 22 – COMPLETING THE SQUARE (a = 1)
Q1 – 12 a = 1, b is even.
Q13 – 24 a = 1, b is either odd, a fraction or a variable.
Write the following quadratic expressions in completed square form.
1. a 2 2k k b 2 2a a c 2 4r r
d 2 4d d e 2 6j j f 2 6v v
g 2 2 1i i h 2 6 1c c i 2 4 3b b
2. a 2 8 7x x b 2 10 5x x c 2 12 9x x
d 2 3x x e 2x x f 2 5x x
g 2 7x x h 2 3x x i 2 9 2x x
3. a 2 11 7x x b 2x bx c 2x bx c
d 2 1
2x x e
2 3
2x x f
2 7
2x x
g 2 1 1
2 4x x h
2 3 5
2 4x x i
2 7 49
2 16x x
EXERCISE 23* – COMPLETING THE SQUARE #2 (a ≠ 1)
Write the following expressions in completed square form as simply as possible.
1. a 22 8x x b 24 20x x c 25 20x x
d 23 12x x e 27 21x x f 2 3x x
g 22 8x x h 26 12x x i 23x x
2. a 22 4 8x x b 25 15 5x x c 23 9 1x x
d 22 3 5x x e 23 6 4x x f 25 3 1x x
g 23 9 2x x h 25 12 3x x i 23 7 2x x
3. a 22 4x x b 22 3 5x x c 22 4 5x x
d 26 3 5x x e 22 3 7x x f 23 2x x
e 27 6 2x x f 27 3 8x x g 25 1 4x x
36 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 22 – COMPLETING THE SQUARE (a = 1) – ANSWERS
1. a (k + 1)2 – 1 b (a – 1)2 – 1 c (r + 2)2 – 4
d (d – 2)2 – 4 e (j + 3)2 – 9 f (v – 3)2 – 9
g (i + 1)2 h (c + 3)2 – 10 i (b – 2)2 – 1
2. a (x + 4)2 – 23 b (q – 5)2 – 20 c (u + 6)2 – 45
d 23 92 4
( )x e 21 12 4
( )x f 25 252 4
( )x
g 27 492 4
( )x h 21 112 4
( )x i 29 732 4
( )x
3. a 2 149112 4
( )x b 22
2 4( )b bx c 22 4
2 4( )b b cx
d 21 14 16
( )x e 23 94 16
( )x f 27 494 16
( )x
g 2 514 16
( )x h 23 114 16
( )x i 274
( )x
EXERCISE 23* – COMPLETING THE SQUARE (a > 1) – ANSWERS
1. a 2
2 2 8x b 2
52
4 25x c 2
5 2 20x
d 2
3 2 12x e 2
3 632 4
7 x f 2
9 34 2
x
g 2
8 2 2x h 2
6 6 1x i 2
1 112 6
3 x
2. a 2
2 1 10x b 2
3 652 4
5 x c 2
3 312 4
3 x
d 2
3 494 8
2 x e 2
3 1 1x f 2
3 2910 20
5 x
g 2
3 352 4
3 x h 2
6 515 5
5 x i 2
7 256 12
3 x
3. a 2
6 2x b 2
49 320 10
5 x c 2
2 1 3x
d 2
5 476 12
3 x e 2
3 4714 28
7 x f 2
2316 12
3 x
g 2
718 4
2 x h 2
543 3
3 x i 2
9 516 8
4 x
37 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 24 – SIMPLIFYING ALGEBRAIC FRACTIONS
Hint: As a general rule, you should always factorise expressions which can be factorised before
doing anything else.
Simplify the following algebraic fractions.
1. 3
3
x 2.
4
2
p 3.
9
6
q 4.
4
16
s
s
5. 2x
x 6.
33
6
t
t 7.
2
3
y
y 8.
4
2
5
10
a
a
9. 3
4
6
8
t
t 10.
s
s 11.
3
2
p
p 12.
3
6
4
n
n
13.
3
2
8
2
p
p
14.
3 6
3
p 15.
4 6
2
y 16. 3
1
x
17. 4
4
s
s
18.
3
6 9q 19.
( 2)( 2)
2
r r
r
20.
( 3)( 4)
4
a a
a
21. 3( 2)
2
i
i
22.
3( 4)
4( 4)
n
n
23.
( 5)( 2)
3 15
b b
b
24.
12( 3)
9( 3)
x
x
25. 12 8
4
c 26.
2 6
3
h
h
27.
2 4
3 6
g
g
28.
6 3
10 5
p
p
29. 2 3e e
e
30.
22 5
2
j j
j
31.
23 7
4
d d
d
32.
1
1
p
p
33.
2 1
1
f
f
34.
6 5
5 6
m
m
35.
12 18
15 10
q r
r q
36.
2
2
2 1
( 1)
k k
k
37. 2
2
3 2
4
m m
m
38.
( 3)( 4)
3 12
q q
q
39.
4 ( 7)
2 ( 3)
x x
x x
40.
2
2
6
2
r r
r r
41. 2
2
3 28
7 12
t t
t t
42.
227 48
9 12
p
p
43.
2
2
2 3 1
2 1
u u
u u
44.
2
2
2 5 12
2 3
n n
n n
45. 2
2
12 10 12
18 15 18
v v
v v
46.
2
2
16 9
9 16
t
t
47.
2
2
25 4
15 4 4
s
s s
48.
3
3
5 15 10
21 7 14
z z
z z
38 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 24 – SIMPLIFYING ALGEBRAIC FRACTIONS – ANSWERS
1. x 2. 2p 3. 3
2
q 4. 1
4
5. x 6. 2
2t 7.
1y
8. 2
2a
9. 34t
10. –1 11. 32
12. 2
3
2n
13. –4p 14. p + 2 15. 2y – 3 16. 3 – x
17. –1 18. 1
2 3q 19. r + 2 20. a + 3
21. 3 22. 34
23. 2
3b 24. 4
3
25. 3c – 2 26. –2 27. 23
28. 35
29. e – 3 30. 12(2 5)j 31.
14(3 7)d 32. –1
33. f + 1 34. –1 35. 65
36. 1
37. 12
mm
38. 3
3
q 39.
2( 7)
3
x
x
40.
31
rr
41. 73
tt
42. 3p + 4 43. 11
uu
44. 41
nn
45. 23
46. –1 47. 5 23 2
ss
48. 57
39 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 25 – ADDING & SUBTRACTING ALGEBRAIC FRACTIONS
Questions 1 – 4 are a little warm up to make sure you remember how to add and subtract
fractions.
Combine and simplify the following.
1. 1 1
3 4 2.
2 3
5 8 3.
5 4
6 9
4. 6 2
11 7 5.
1 2
b b 6.
4 1
q q
7. 2 2
1 5
f f 8.
7 3
1 1h h
9. 2 2
4y y
c c
10. 2 3
3 2j j
11. 3 1
5 4m m
12. 2
3 4
x x
y y
13. 2 3 4
5 10 15e e e 14.
1 1
1 2a a
15.
1 1
1 1p p
16. 2 3
2 1s s
17.
4 2
3 1g g
18.
2 2
1 3d d
19. 3
2 2
i i
i i
20. 2
3 4
( 4) 4k k
21. 2
4
( 3) 3
n n
n n
22. 2
2 2
3 2 2q q q
23.
7 3
2 (2 )(3 )s s s
24. 2 2
1 1
12 2 8u u u u
40 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 25 – ADDING AND SUBTRACTING ALGEBRAIC FRACTIONS – ANSWERS
1. 7
12 2.
3140
3. 7
18 4.
2077
5. 3b
6. 3q
7. 2
6
f 8.
41h
9. 2
3 y
c 10.
136 j
11. 7
20m 12.
512
xy
13. 1330e
14. 2 3
1 2
a
a a
15. 2
2
1 p 16.
5 8
2 1
s
s s
17.
2 5
3 1
g
g g
18.
4
1 3d d 19.
2
2 4
4
i i
i
20.
2
4 19
4
k
k
21.
2
1
3
n n
n
22.
2
2 1
q
q q
23.
24 7
2 3
s
s s
24.
2 5
4 3 2
u
u u u
41 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 26 – MULTIPLYING & DIVIDING ALGEBRAIC FRACTIONS
Hint: Remember that x + 2 is NOT divisible by either 2 or x.
Combine and simplify the following.
1. 2 3
x x 2.
2
3 2
b b 3.
3 4
2 9
p q
4. 2
4
f
f 5.
15
5 2
n
n 6.
3 5
10 2
c
c
7.
2 6
3
q
q 8.
4 6
3
a j
j a 9.
1 6
3 1
r
r
10. 2 3
3
e e
e e
11.
2
2
3 2 2
( 1)
y y y
y y
12.
3 6
3 2 6
d
d
13. 2 4
u u 14.
1 3
s s 15.
2 6
3g g
16. 3 3xy x
x y 17.
3 2
2 3 6
h
h h
18.
3 2
2 2
3 2
( 5) 7 10
m m
m m m
19. 2 8 4
3 3
k k
k k
20.
2 4 2
2 3
x x
x
21.
2( 4) 4
2 6
t t
t t
22.
2
2
2 3 2 3
3 3 7 6
y y y
y y y
23. 2
3 4 9 12
1 ( 1)
i i
i i
24.
2 2
2 2
4 3 6
3 4 6 8
a a a a
a a a a
42 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 26 – MULTIPLYING AND DIVIDING ALGEBRAIC FRACTIONS
1. 2
6x
2. 2
3b
3. 2
3
pq
4. 12
5. 32
6. 34
7. 2q 8. 8 9. 2
10. 2 33
ee
11. 2( 2)
1
y
y
12. 1
13. 2 14. 13
15. 19
16. 2y
x
17. 92h
18. 3 ( 2)
2( 5)
m m
m
19. 2 20. 3 21. 3(t – 4)
22. 3 2
1
y
y
23. 13
i
24. 1
43 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 27 – MIXED ALGEBRAIC FRACTIONS PROBLEMS
Simplify the following expressions.
1. 2
3 1
2i i i
2.
22
6
d
d 3. 2
3 4
2 ( 2)r r
4. 2
4 2
4 2h h
5.
2
2
6
2 6
s s
s s
6.
2 12
3 2 4
f
f
7.
2
2
3 9
4 2
e e
y y 8. 2
3 3
4 7 12
c c
c c c
9. 2
2
3 2 2
q
q q q
10. 2 1
1 2t t
11.
2
2
6 12
10 11 6
y y
y y
12.
2 2
( 2)( 2) 2
b
b b b
13. 3 2
( 3)( 2) 3
s
s s s
14. 2
1 4
( 3) 3a a
15.
6 6 15
2 5 3
g g
g g
16. 5 2
( 1)( 1) 1k k k
17.
2 3
( 4)( 3) 3j j j
18.
3 24
3 4 9 12
n n
n n
19. 2 24 9
3 2
r r
r r
20.
5 2
( 2)( 3) 3m m m
21. 2
7 2 2
6 2
p
p p p
22. 2
2 2
3 5 6
t t
t t t
23. 2 2
2 1
2 6 2 3n n n n
24. 2
2
5 6 2 6
x x
x x x
44 MANIPULATING ALGEBRAIC EXPRESSIONS
EXERCISE 27 – MIXED EXERCISE ON ALGEBRAIC FRACTIONS – ANSWERS
1. 1( 2)
ii i
2.
3d
3.
2
3 10
( 2)
r
r
4. 22h
5. 32 3ss
6. 2
7.
6
ey
8. 1 9. 11q
10. 3 5( 1)( 2)
tt t
11. 3 4
5 2
y
y
12. 2
4
4b
13. 22( 3)( 2
4 3)
ss s
s
14. 2
4 11
( 3)
a
a
15. 6
16.
2
2
1
3k
k
17. 3 14
( 4)( 3)
j
j j
18. 34n
19. ( 2)( 3)r r 20. 1 2( 2)( 3)
mm m
21. 5 4
( 2)( 3)
p
p p
22. 1 23. 3 4(2 3)( 2)( 1)
nn n n
24. 2
2( 2)(43
4)
xx x
x